The modelling of the interactions between a wave and an obstacle receiving this wave, such as a target placed in the responsive zone of a sensor, finds an advantageous application in nondestructive testing.
A method of modelling called “finite elements” is known consisting in applying a tiling of the three-dimensional space surrounding the obstacle and in evaluating the aforesaid interactions for all the tiles of the space.
Methods of computation by “finite elements” afford a solution to a problem posed in the form of partial differential equations. They are based on a representation of the space under study by an assemblage of finite elements, inside which are defined approximation functions determined in terms of nodal values of the physical quantity sought. The continuous physical problem therefore becomes a discrete finite element problem where the nodal quantities are the new unknowns. Such methods therefore seek to approximate the global solution, rather than the starting equations in the partial spatial derivatives.
The discretization of the space taken into account ensures that the latter is entirely covered by finite elements (lines, surfaces or volumes), this operation is called “meshing” in two dimensional space (2D) or “tiling” in three-dimensional space (3D). The elements involved are either rectangular or triangular in 2D, or parallelepipedal or tetrahedral in 3D. They may be of different sizes, distributed uniformly or otherwise over the surface.
In general, the physical quantity sought, such as an electrostatic potential or a pressure value, is known on the boundary of the domain. This boundary may be fictitious. Boundary conditions are imposed there. The potential is therefore unknown inside the same domain. A node is then defined as being a vertex of an element. The unknowns of the problem are therefore the values of the potential at each node of the domain as a whole.
By way of illustration, FIG. 6 of the prior art represents an exemplary surface, consisting of two materials M1 and M2, of different electromagnetic properties, and meshed by triangular elements each comprising three nodes Ai, Bi and Ci. The domain as a whole is delimited by a boundary F.
Once the mesh has been defined, several approaches exist for transforming the physical formulation of the problem into a discrete modelling by finite elements. If the problem is formulated through differential equations and consists in minimizing a functional, then a variational procedure is generally applied. This transformation leads to a matrix formulation which when solved gives the nodal solutions, the solutions at the non-nodal points being obtained by linear interpolation.
Nevertheless, such computations, in three dimensions, require considerable computing resources and generate very long computation times, despite the enhancement in the performance of software allowing the implementation of these computations.
Admittedly, 2D problems, often simplified by symmetry conditions that are advantageous for modelling only part of the geometry, are solved rapidly. However, this is not so for 3D problems, which are the most frequent. FIG. 6 shows how the fineness of the mesh, that is to say the ratio of the size of an element to that of the smallest detail of the domain, amplifies the number of nodes.
Consequently, the number of equations and of unknowns increases proportionally, and, hence, the computation time required for solving the problem. It is important to point out that the generation of the mesh, namely the discretization of the workspace, and the generation of the list of nodes consumes greater computation time than that required for solving the problem.
The present invention aims to improve the situation.